Physics 207, Lecture 23, Nov. 22

1. Physics 207, Lecture 23, Nov. 22 • Agenda: Catch up • Chapter 18, Superposition and Standing Waves • Superposition • Interference • Standing Waves • Nodes, Anti-nodes Assignments: • Problem Set 9 due Tuesday, Dec. 5, 11:59 PM Ch. 18: 3, 18, 30, 40, 58 • Mid-term 3, Tuesday, Nov. 28, Chapters 14-17, 90 minutes, 7:15-8:45 PM in rooms 105 and 113 Psychology • Monday is a review for Tuesday’s mid-term

2. Superposition & Interference(How do waves add) • Consider two harmonic waves A andB meeting. • Same frequency and amplitudes, but phases differ (f). • The displacement versus time for each is shown below: A(t) B(t) What does C(t) =A(t)+ B(t) look like ? Wave Superposition

3. Superposition & Interference • Consider A + B, A(x,t)=A cos(kx–wt)B(x,t)=A cos(kx–wt+f) • We can show:C = 2A cos(f/2) cos(kx– wt+f/2) • Using half-angle identities……see text 18.1 A(t) B(t) f Amplitude = 2A cos (f/2) C(kx-wt) Phase shift = f / 2

4. Lecture 23, Exercise 1Superposition • Two continuous harmonic waves with the same frequency and amplitude but, at a certain time, have a phase difference of 170° are superimposed. Which of the following best represents the resultant wave at this moment? Original wave (the other has a different phase) (A) (B) (D) (C) (E)

5. Superposition & Interference • We have just seen that when waves combine (superimpose) the result can either be bigger or smaller than the original waves. • Waves can add “constructively” or “destructively” depending on the relative sign of each wave. will add constructively will add destructively • In general, both may happen Pulse Superposition

6. DESTRUCTIVEINTERFERENCE CONSTRUCTIVEINTERFERENCE Superposition & Interference • Consider two harmonic waves A and B meet at t=0. • They have same amplitudes and phase, but 2 = 1.15 x 1. • The displacement versus time for each is shown below: Beat Superposition A(1t) B(2t) C(t) =A(t)+B(t)

7. Aside: Why superposition works • The equation governing waves (Chapter 16, “the Wave Equation”) is linear. For linear equations, if we have two (or more) separate solutions, f1 and f2, then B f1 + C f2 is also a solution . • For linear equations, if we have two (or more) separate solutions, f1 and f2, then B f1 + C f2 is also a solution • This is called the “Superposition Principle” • You have already seen this in the case of simple harmonic motion: linear in x ! x(t) = B sin(t)+C cos(t)

8. Superposition & Interference • Consider A + B, yA(x,t)=A cos(k1x–w1t)yB(x,t)=A cos(k2x–w2t) And let x=0, y=yA+yB = 2A cos[2p (f1 – f2)t/2] cos[2p (f1 + f2)t/2] and |f1 – f2| ≡ fbeat = = 1 / Tbeat A(1t) B(2t) t Tbeat C(t)=A(t)+B(t)

9. Lecture 23, Exercise 2Superposition • The traces below show beats that occur when two different pairs of waves are added (the time axes are the same). • For which of the two is the difference in frequency of the original waves greater? Pair 1 Pair 2 The frequency difference was the samefor both pairs of waves. Need more information.

10. Interference of Waves • A path corresponds to a phase [recall the cos(2px/l)] Path (or x)  phase = 2p (path/ l) (modulo 2p ) • If two waves start out “in-phase” (at the same time) and then travel different distances before they are superimposed then the path difference, DL, corresponds to a phase difference with constructive or destructive interference. with n = 0, 1, 2, …

11. Interference of Waves • 2D Surface Waves on Water In phase sources separated by a distance d d

12. Interference of Sound Sound waves interfere, just like transverse waves do. The resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with. Constructive interference: Destructive interference:

13. Interference of Waves, Splitting and Guiding • Controlling wave sources is exploited in numerous applications Optical Y Splitter A Crystal with Line Defect Acting as a Waveguide: Si (n=3.4); Period A = 0.58mm; Filling Factor = 5/16; Excitation l = 1.55mm Light turning a corner

14. Lecture 23, Example Interference • A speaker sits on a pedestal 2 m tall and emits a sine wave at 343 Hz (the speed of sound in air is 343 m/s, so l = 1m ). Only the direct sound wave and that which reflects off the ground at a position half-way between the speaker and the person (also 2 m tall) makes it to the persons ear. • How close to the speaker can the person stand (A to D) so they hear a maximum sound intensity assuming there is no phase change at the ground (this is a bad assumption)? t1 t0 d t0 D h A A B C The distances AD and BCD have equal transit times so the sound waves will be in phase. The only need is for AB = 1 wavelength

15. Lecture 23, Example Interference • The geometry dictates everything else. AB = l AD = BC+CD = BC + (h2 + (d/2)2)½ = d AC = AB+BC = l +BC = (h2 + d/22)½ Eliminating BC gives l+d = 2 (h2 + d2/4)½ l + 2ld + d2 = 4 h2 + d2 1 + 2d = 4 h2 / l d = 2 h2 / l – ½ = 7.5 m t1 t0 7.5 t0 D A A 4.25 3.25 B C Because the ground is more dense than air there will be a phase change of p and so we really should set AB to l/2 or 0.5 m.

16. Main point • Path differences will give phase differences. • This will lead to a superposition with constructive or destructive interference. • If two waves start out “in-phase” (at the same time) and then travel different distances before they are superimposed then the path difference, DL, corresponds to a phase difference with:

17. Standing Waves: A special kind of superposition • Consider A + B, same l and w but traveling to the left and right.A(x,t)=A cos(kx–wt)B(x,t)=A cos(kx+wt+p) Now C(x,t) = 2A cos(2px/l) cos(wt)and there is no net energy flow. If f = p/2 then C’(x,t) = 2A sin(2px/l) sin(wt) These are “standing waves”. • This describes motion on a bound string (length L) C(0,t) = C(L,t) = 0 if L = n l/2  l = 2 L/n • Or more generally C’(x,t) = 2A sin(p n x/L) sin(wt) n =1 n =2 n =3 n =4

18. Guitar Strings A combination wave composed of the 1st harmonic and the third harmonic.

19. What makes instruments unique is the combination of harmonics produced by the different instruments. Flutes produce primarily the 1st harmonic They have a very pure tone Oboes produce a broad range of harmonics and sound very different Music

20. Combining Waves Revisited

21. Combining Waves Fourier Synthesis

22. Three ways to make sound Vibrate a string Vibrate an air column Vibrate a membrane Musical Instruments

23. Violin, viola, cello, string bass Guitars Ukuleles Mandolins Banjos All vibrate a structure to “amplify” the sound Vibrating Strings Vibrating Air Columns • Pipe Organs • Brass Instruments • Woodwinds • Whistles Vibrating Membranes • Percussion Instruments • Drums • Bongos

24. Standing Waves in Pipes Open at one end: Pressure AntiNode at closed end Displacement Node at closed end l = 4 L / n n = 1,3,5… Open at both ends: Pressure(speed) Node at ends Displacement AntiNode at ends l = 2 L / nn = 1,2,3..

25. Organ Pipe Example A 0.9 m organ pipe (open at both ends) is measured to have it’s first harmonic (i.e., its fundamental) at a frequency of 382 Hz. What is the speed of sound (refers to energy transfer) in the pipe? L=0.9 m f = 382 Hzandf l = vwith l = 2 L / n(n = 1) v = 382 x 2(0.9) m  v = 687 m/s

26. Lecture 23, Exercise 3Standing Waves • What happens to the fundamental frequency of a pipe, if the air (v =300 m/s) is replaced by helium (v = 900 m/s)? Recall: f l = v (A) Increases (B) Same (C) Decreases

27. Recap, Lecture 23 • Agenda: Catch up • Chapter 18, Superposition and Standing Waves • Superposition • Interference • Standing Waves • Nodes, Anti-nodes Assignments: • Problem Set 9 due Tuesday, Dec. 5, 11:59 PM Ch. 18: 9, 17, 21, 39, 53a (tentative) • Mid-term 3, Tuesday, Nov. 28, Chapters 14-17, 90 minutes, 7:15-8:45 PM in rooms 105 and 113 Psychology • Monday is a review session for Tuesday’s mid-term • Have a good Thanksgiving holiday and see you Monday!